Some Classical and Quantum Algebras

TL;DR

This paper explores classical and quantum algebraic structures, including BV algebras, quantum operator algebras, and vertex operator algebras, highlighting their interrelations and deformations in mathematical physics.

Contribution

It introduces quantum operator algebras as a generalization of classical operator algebras and connects BV algebras with vertex operator algebras.

Findings

QOAs as deformations of commutative algebras

VOAs as special cases of QOAs with additional structures

Established a link between BV algebras and VOAs

Abstract

We discuss the notion of a Batalin-Vilkovisky (BV) algebra and give several classical examples from differential geometry and Lie theory. We introduce the notion of a quantum operator algebra (QOA) as a generalization of a classical operator algebra. In some examples, we view a QOA as a deformation of a commutative algebra. We then review the notion of a vertex operator algebra (VOA) and show that a vertex operator algebra is a QOA with some additional structures. Finally, we establish a connection between BV algebras and VOAs.